Assessment of metrical property of images space system observation

The article deals with mathematical model of residual errors, arising at the space images processing, received through the space optical and electronic observation systems. From the analysis of conditions of their emerging it was found, that errors in georeferencing and in the interior geometry of the space image are the basic components of this model. Considering the basic regulations known through theorems by Kotelnikov and Shannon represented in the signal theory, borders of confidence interval of measurements errors digital image were defined. Basing upon ascertained type of mathematical model of residual errors the final accuracy of processing modern space images was forecast. The predicted values’ errors at creating final cartographic products of processing home- and foreign made space photographs were tabulated. The assessment of their indicators was made. The principal causes of the arising divergence between theoretical estimates of the accuracy of measuring the digital images and the practical results were defined. Recommendation for improving the metrological provision of the Russian data was given.

Metrical property for GCFϵ expansion with the parameter function ϵ(k) = c(k + 1)

This paper is concerned with the metric properties of the generalized continued fractions (GCFϵ) with the parameter function ϵ(kn), where kn is the nth partial quotient of the GCFϵ expansion. When -1 < ϵ(kn) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory128 (2008) 1506–1515] obtained the following metrical properties: [Formula: see text] which are entirely unrelated to the choice of ϵ(kn) ∈ (-1, 1]. Here we deal with the case of ϵ(k) = c(k + 1) with constant c ∈ (0, ∞). It is proved that: [Formula: see text] which change with the real c ∈ (0, ∞). Note that [Formula: see text] as c → 0, it indicates that when c → 0, the GCFϵ has the same metrical property as the case of -1 < ϵ(kn) ≤ 1.

Poincaré Theta Series and Singular Sets of Schottky Groups

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.